#### Dark energy domination in the Virgocentric flow

A&A
Astronomy & Astrophysics
A. D. Chernin 1 2
I. D. Karachentsev 0
O. G. Nasonova 0
P. Teerikorpi 2
M. J. Valtonen 2
V. P. Dolgachev 1
L. M. Domozhilova 1
G. G. Byrd 3
0 Special Astrophysical Observatory , Nizhnii Arkhys 369167 , Russia
1 Sternberg Astronomical Institute, Moscow University , Moscow 119899 , Russia
2 Tuorla Observatory, Department of Physics and Astronomy, University of Turku , 21500 Piikkiö , Finland
3 University of Alabama , Tuscaloosa, AL 35487-0324 , USA
Context. The standard ΛCDM cosmological model implies that all celestial bodies are embedded in a perfectly uniform dark energy background, represented by Einstein's cosmological constant, and experience its repulsive antigravity action. Aims. Can dark energy have strong dynamical effects on small cosmic scales as well as globally? Continuing our efforts to clarify this question, we now focus on the Virgo Cluster and the flow of expansion around it. Methods. We interpret the Hubble diagram from a new database of velocities and distances of galaxies in the cluster and its environment, using a nonlinear analytical model, which incorporates the antigravity force in terms of Newtonian mechanics. The key parameter is the zero-gravity radius, the distance at which gravity and antigravity are in balance. Results. 1. The interplay between the gravity of the cluster and the antigravity of the dark energy background determines the kinematical structure of the system and controls its evolution. 2. The gravity dominates the quasi-stationary bound cluster, while the antigravity controls the Virgocentric flow, bringing order and regularity to the flow, which reaches linearity and the global Hubble rate at distances >15 Mpc. 3. The cluster and the flow form a system similar to the Local Group and its outflow. In the velocity-distance ∼ diagram, the cluster-flow structure reproduces the group-flow structure with a scaling factor of about 10; the zero-gravity radius for the cluster system is also 10 times larger. Conclusions. The phase and dynamical similarity of the systems on the scales of 1−30 Mpc suggests that a two-component pattern may be universal for groups and clusters: a quasi-stationary bound central component and an expanding outflow around it, caused by the nonlinear gravity-antigravity interplay with the dark energy dominating in the flow component.
Local Group - galaxies; clusters; individual; Virgo cluster - dark matter - dark energy
1. Introduction
Early studies of the motion of our Galaxy
(reviewed by Huchra
1988)
led to the discovery of the peculiar velocity of the Local
Group towards the Virgo cluster. To illustrate this, imagine
the distance to Virgo to be 17 Mpc and the Hubble constant
H0 = 72 km s−1 Mpc−1, then the expected cosmological
velocity of the cluster is 1224 km s−1. The observed velocity is
about 1000 km s−1. The difference reflects our peculiar velocity
of about 220 km s−1 towards Virgo.
This estimate of the retarded expansion (the so-called “Virgo
infall”) views the universal expansion as existing not only on
“truly cosmological” scales ∼1000 Mpc, but also on local scales
of ∼10 Mpc only weakly disturbed. Observational and
theoretical aspects of the Virgo infall were studied e.g. by Silk (1974,
1977),
Peebles (1976)
,
Hoffman & Salpeter (1982)
,
Sandage
(1986)
,
Teerikorpi et al. (1992)
, and
Ekholm et al. (1999
, 2000).
1.1. The Virgocentric flow
The Virgo infall is a particular feature of what is known as
the Virgocentric flow: hundreds of galaxies (our Galaxy among
them) are receeding away from the Virgo cluster. Recent data
(Karachentsev & Nasonova 2010)
show that the flow
velocities range from nearly zero around 5−8 Mpc from the cluster
center to about 2000 km s−1 at the distances of about 30 Mpc.
For R > 10−15 Mpc, a roughly linear velocity-distance relation
may be seen in the flow, making it resemble the global Hubble
expansion. But strong deviations from the linear relation exist at
smaller distances, not surprisingly: 1) the linear expansion flow,
V = HR, is the property of the universe beyond the cosmic “cell
of uniformity” (≥300 Mpc); 2) the Virgo cluster is a strong
local overdensity; 3) the matter distribution around the cluster is
highly nonuniform up to the distance of about 30 Mpc.
Besides the natural deviations, we have here an example of a
paradox realized by Sandage (1986, 1999). He saw a mystery in
the fact that the local expansion proceeds in a quite regular way.
Moreover, the rate of expansion is similar, if not identical, to the
global Hubble constant
(Ekholm et al. 2001; Thim et al. 2003;
Karachentsev 2005; Karachentsev et al. 2002a, 2003b; Whiting
2003)
. Originally, the Friedmann theory described the
expansion of a smooth, uniform self-gravitating medium. But the
linear cosmological expansion was discovered by Hubble where it
should not be: in the lumpy environment at distances <20 Mpc.
1.2. The local relevance of dark energy
Soon after the discovery of dark energy
(Riess et al. 1998;
Perlmutter et al. 1999)
on global scales, we suggested
(Chernin
et al. 2000; Chernin 2001; Baryshev et al. 2001; Karachentsev
et al. 2003a; Chernin et al. 2004; Teerikorpi et al. 2006)
that
dark energy with its omnipresent and uniform density
(represented by Einstein’s cosmological constant) may provide the
dynamical background for a regular quiescent expansion on local
scales, resolving the Hubble-Sandage paradox. Our key
argument came from the fact that the antigravity produced by dark
energy is stronger than the gravity of the Local Group at
distances larger than about 1.5 Mpc from the group center1.
The present paper extends our work from groups to the scale
of clusters, focusing on the Virgo cluster.
2. The phase space structure
Recent observations of the Virgo cluster and its vicinity permit a
better understanding of the physics behind the visible structure
and kinematics of the system.
2.1. The dataset
The most complete list of data on the Virgo Cluster and
the Virgocentric flow has been collected by
Karachentsev &
Nasonova (2010)
. These include distance moduli of
galaxies from the Catalogue of the Neighbouring Galaxies
(=CNG,
Karachentsev et al. 2005)
and also from the literature with the
best measurements prefered. Distances from the tip of the red
giant branch (TRGB) and the Cepheids are used from the CNG
together with new TRGB distances
(Karachentsev et al. 2006;
Makarov et al. 2006; Mei et al. 2007)
. For galaxy images in
two or more photometric bands obtained with WFPC2 or ACS
cameras at the Hubble Space Telescope, the TRGB method
yields distances with an accuracy of about 7%
(Rizzi et al.
2007)
. The database includes also data on 300 E and S0 galaxies
from the surface brightness fluctuation (SBF) method by
Tonry
et al. (2000)
with a typical distance error of 12%. The total
sample contains the velocities and distances of 1371 galaxies
within 30 Mpc from the Virgo cluster center. Especially
interesting is the sample of 761 galaxies selected to avoid the
effect of unknown tangential (to the line of sight) velocity
components. The velocity-distance diagram for this sample taken from
Karachentsev & Nasonova (2010)
is given in Fig. 1.
2.2. The zero-velocity radius and the cluster mass
The zero-velocity radius within the retarded expansion field
around a point-like mass concentration means the distance where
the radial velocity relative to the concentration is zero. In the
ideal case of the mass concentration at rest within the expanding
Friedmann universe this is the distance where the radial
peculiar velocity towards the concentration is equal to the Hubble
velocity for the same distance. Using Tully-Fisher distances in
the Hubble diagram,
Teerikorpi et al. (1992)
could for the first
time see the location of the zero-velocity radius R0 for the Virgo
system, so that R0/RVirgo ≈ 0.45 or R0 ≈ 7.4 Mpc. The work by
Karachentsev & Nasonova (2010)
puts the zero-velocity radius
at R0 = 5.0−7.5 Mpc. For R < R0, positive and negative
velocities appear in practically equal numbers; for R > R0, the
velocities are positive with a few exceptions likely due to errors in
1 The increasing observational evidence and theoretical considerations
favoring this view have been discussed by
Macciò et al. (2005)
,
Sandage
(2006)
, Teerikorpi et al. (2006, 2008), Chernin et al.
(2006, 2007a−c)
,
Byrd et al. (2007),
Valtonen et al. (2008)
,
Balaguera-Antolinez et al.
(2007)
,
Bambi (2007)
,
Chernin (2008)
,
Niemi & Valtonen (2009)
,
Guo
& Shan (2009)
. For some counter-arguments to this new approach see
Hoffman et al. (2008)
and
Martinez-Valquero et al. (2009
).
Fig. 1. Hubble diagram with Virgocentric velocities and distances for
761 galaxies of the Virgo Cluster and the outflow
(Karachentsev &
Nasonova 2010)
. The two-component pattern is outlined by bold dashed
lines. The zero-gravity radius RZG ∼ 10 Mpc is located in the zone
between the components. The broken line is the running median used
by K & N to find the zero-velocity radius R0 = 6−7 Mpc. Two lines
from the origin indicate the Hubble ratios H = 72 km s−1 Mpc−1 and
HV = 65 km s−1 Mpc−1. Two curves show special trajectories,
corresponding to the parabolic motion (E = 0; the upper one) and to the
minimal escape velocity from the cluster potential well (the lower one).
The flow galaxies with the most accurate distances and velocities (big
dots) occupy the area between these curves.
distances. The zero-velocity radius gives the upper limit of the
size of the gravitationally bound cluster, and the diagram shows
that the Virgocentric flow starts at R ≥ R0 and extends at least
up to 30 Mpc.
The zero-velocity radius has been often used for estimating
the total mass M0 of a gravitationally bound system. According
to
Lynden-Bell (1981)
and
Sandage (1986)
, the spherical model
with Λ = 0 leads to the estimator
M0 = (π2/8G)tU−2R30.
With the age of the universe tU = 13.7 Gyr
(Spergel et al. 2007)
,
Karachentsev & Nasonova (2010)
find the Virgo cluster mass
M0 = (6.3 ± 2.0) × 1014 M . This result agrees with the virial
mass Mvir =46××1011401M4 M estimated by
Hoffman & Salpeter
(1982)
and ∼ of
Valtonen et al. (1985)
and
Saarinen
& Valtonen (1985)
.
Teerikorpi et al. (1992)
and
Ekholm et al.
(1999
, 2000) found that the real cluster mass might be from 1
to 2 the virial mass, or (0.6−1.2) × 1015 M .
Tully & Mohayaee
(2004)
derived the mass 1.2 × 1015 M with the “numerical
action” method.
(1)
2.3. The Virgo infall
Figure 1 clearly shows the Virgo infall: at distances R > R0,
the galaxies are located mostly below the Hubble line VH = HR
with H = 72 km s−1 Mpc−1
(Spergel et al. 2007)
. The deviation
of the median velocity of the flow Vm from the Hubble velocity
is well seen in the distance range 7−13 Mpc. At larger distances
the retardation effect gets weaker and gradually sinks below the
measurement error level.
For the distance and peculiar velocity of the Milky Way, we
take the figures from
Karachentsev & Nasonova (2010)
: the
distance to the Virgo cluster 17 Mpc, the recession velocity 1004 ±
70 km s−1; with the Hubble constant H = 72 km s−1 Mpc−1 the
regular expansion velocity is 1224 km s−1. Then the peculiar
velocity directed to the cluster center δV = 220 ± 70 km s−1
(this
agrees with the result based on TF distances and a different
analysis by Theureau et al. 1997; a complete velocity field on the
scale of 10−20 Mpc is given by Tully et al. 2008)
. The effect is
not very strong. It is much more instructive, however, that there
are big nonlinear deviations from the linear velocity-distance law
near the cluster: the infall effect is strongest there.
2.4. Two-component phase structure
The Virgo system reveals an obvious two-component structure
in Fig. 1: the cluster and the flow, outlined roughly with bold
dashed lines. As we noted, the positive and negative velocities
(from −2000 to +1700 km s−1) are seen in the cluster area in
equal numbers, so the component is rather symmetrical to the
horizontal line V = 0. The border zone between the
components, in the range 6 < R < 12 Mpc, is poorly populated, and
the velocities are considerably less scattered here, from −400
to +600 km s−1. This zone contains the zero-velocity radius. The
flow component with positive velocities is at distances >12 Mpc.
It is rather symmetrical to the line V = HR. The velocities are
scattered within ±1000 km s−1 around the symmetry line.
A much lower velocity dispersion is seen in the more
accurate data on 75 galaxies with TRGB and Cepheid distances
(mostly from CNG) located from 10 to 25 Mpc from the cluster
center. In the diagram of Fig. 1, the subsample (dots) occupies
a strip less than 500 km s−1 wide. They are scattered with the
mean dispersion of about 250 km s−1 around the line V = HV R,
where HV 65 km s−1 Mpc−1. This line is the median line for
the subsample.
According to the simplest, most straightforward and quite likely
view adopted in the ΛCDM cosmology, dark energy is
represented by Einstein’s cosmological constant Λ, so that ρV =
2.5. Similarity with the Local system (c2/8πG)Λ. If so, then dark energy is the energy of the cosmic
vacuum
(Gliner 1965)
and is described macroscopically as a
perThe two-component phase structure of the Virgo system looks fectly uniform fluid with the equation of state pV = −ρV (here pV
very similar to that of the Local system (the Local Group and is the dark energy pressure; c = 1). This interpretation implies
the outflow). The diagram of Fig. 2
(Karachentsev et al. 2009)
that dark energy exists everywhere in space with the same
denis based on the most complete and accurate data obtained with sity and pressure.
the TRGB method using the HST. The velocities within the Local dynamical effects of dark energy have been studied
Local Group range from −150 to +160 km s−1, about 10 times by Chernin et al. (2000, 2009),
Chernin (2001)
, Baryshev et al.
less than the velocity spread in the Virgo cluster. The zone be- (2001), Karachentsev et al. (2003a),
Byrd et al. (2007)
, and
tween the two components of the Local system is in the range
Teerikorpi et al. (2008)
. For the Local Group and the outflow
0.7−1.2 Mpc, also 10 times less than in Fig. 1. The local flow close to it, we showed that the antigravity produced by the dark
has a dispersion of about 30 km s−1 around the line V = HLR, energy background exceeds the gravity of the group at distances
where HL = HV 60 km s−1 Mpc−1. This line defines the me- beyond 1.5 Mpc from the group barycenter. Similar results
dian for the local flow. The dispersion is about 10 times less than were obtained for the nearby M 81 and Cen A groups
(Chernin
in the Vigocentric flow for the subsample of the accurate dis- et al. 2007a,b)
.
tances. The local phase diagram of Fig. 2, if zoomed by a factor Extending now our approach to the cluster scale, we consider
of 10, would roughly reproduce the phase structure of the Virgo the Virgo system (the cluster and the flow) embedded in the
unisystem. We will see that the phase similarity is also supported form dark energy background. We describe the system – in the
by the dynamical similarity. first approximation – in a framework of a spherical model
(Silk
1974, 1977; Peebles 1976)
where the cluster is considered as
a spherical gravitationally bound quasi-stationary system. The
3. Gravity-antigravity interplay galaxy distribution in the flow is represented by a continuous
Dark energy is a form of cosmic energy that produces antigrav- dust-like (pressureless) matter. The flow is expanding, and it is
ity and causes the global cosmological acceleration discovered assumed that the concentric shells of matter do not intersect each
by Riess et al. (1998) and
Perlmutter et al. (1999)
in observa- other in their motion (the mass within each shell keeps constant).
tions of SNe type Ia at horizon-size distances of about 1000 Mpc. Because the velocities in the cluster and the flow are much
These and other observations, especially the cosmic microwave less than the speed of light, we may analyze the dynamics of
background (CMB) anisotropies
(Spergel et al. 2007)
, indicate the cluster-flow system in terms of Newtonian forces. A
spherthat the global dark energy density, ρV = (0.73 ± 0.03) × ical shell with a Lagrangian coordinate r and a Eulerian
dis10−29 g cm−3, makes up nearly 3/4 of the total energy of the uni- tance R(r, t) from the cluster center is affected by the
gravitaverse. tion of the cluster and the flow galaxies within the sphere of the
Fig. 2. Hubble diagram for 57 galaxies of the Local Group and the local
flow
(Karachentsev et al. 2009)
. The thin line indicates the zero-velocity
radius R0 = 0.7−0.9 Mpc. The two-component phase structure is
outlined by the bold dashed line. Zoomed in about 10 times, the
groupscale structure well reproduces the cluster-scale structure (Fig. 1). The
zero-gravity radius RZG is about 10 times smaller for the group than
the same key quantity of the cluster system. The similarity supports a
universal cosmic two-component grand-design for groups and clusters.
3.1. Antigravity in Newtonian description
in the reference frame related to the cluster center. Here M(r) is
the total mass within the radius R(r, t).
The flow is also affected by the repulsion produced by the
local dark energy. The resulting acceleration in Newtonian terms
is given by the “Einstein antigravity law”:
(8)
(9)
Karachentsev & Nasonova (2010)
argue in favor of one virial
mass for the cluster mass; if that is the case, the lower limit for
RZG in Eq. (7) may be more realistic.
Equations (6) and (7) indicate that the zero-gravity
radius RZG of the Virgo system is not far from the zero-velocity
radius R0 = 5.0−7.5 Mpc. Indeed, one expects that RZG is slightly
larger than R0
(Teerikorpi et al. 2008)
in the standard
cosmology. The zero-velocity radius constrains the size of the cluster
(Sect. 2), and there gravity also dominates (R < R0 ∼< RZG), the
necessary condition for a gravitationally bound cluster. On the
other hand, the Virgocentric flow is located beyond the distance
RZG R0, hence the flow is dominated dynamically by the dark
energy antigravity.
The zero-gravity radius of the Virgo system is about 10 times
larger than that of the Local system for which RZG = 1−1.3 Mpc
(Chernin 2001; Chernin et al. 2000, 2009)
. We see here the same
factor 10 as for the phase structures of the Virgo and Local
systems (Sect. 2). Thus the phase similarity is complemented by the
dynamical similarity.
3.3. Antigravity push
Now we return to the Milky Way and estimate the net
gravityantigravity effect at its distance RMW = 17 Mpc from the cluster
center. With RZG = 9−11 Mpc, we find that here the antigravity
force is stronger than the gravity:
FE/|FN|
4 − 7, R = RMW.
As we see, the extra gravity produced by the Virgo cluster mass
is actually overbalanced by the antigravity of the dark energy
located within the sphere of the radius RMW around the
cluster. Thus the Milky Way experiences not a weak extra pull
(cf. Sects. 1 and 2), but rather a strong push (in the Virgocentric
reference frame) from the Virgo direction.
The dark energy domination in the flow changes the current
dynamical situation drastically in comparison with the earlier
models by Silk (1974, 1977) and
Peebles (1976)
. At the
periphery of the Virgocentric flow (at about 30 Mpc) the force ratio
is impressively high: FE /|FN | 20−37.
3.4. Antigravity domination
The basic trend of the flow evolution is seen from Eq. (5). With
M(r) = M = const., it takes at prolonged times the form
V(r, t = ∞) =
8π
3
1/2
GρΛ
R.
(3)
(4)
(5)
43π R(r, t)3
FE(r, t) = +G 2ρΛ R(r, t)2
8π
3
= +
GρΛR(r, t).
Here −2ρΛ = ρΛ +3pΛ is the effective (General Relativity)
gravitating density of dark energy
(for details see, e.g., Chernin 2001)
.
It is negative, and therefore the acceleration is positive, speeding
up the shell away from the center.
3.2. Zero-gravity radius in the point-mass approximation
The equation of motion of an individual shell has the form
GM(r)
R¨(r, t) = FN + FE = − R(r, t)2 + 3
8π
GρΛR(r, t).
Its integration leads to the mechanical energy conservation law
for each matter shell:
1
2
V(r, t)2 =
GM(r)
R(r, t)
Here V(r, t) = R˙ (r, t) is the Eulerian radial velocity of the shell,
and E(r) is its total mechanical energy (per unit mass). This is the
basic equation which describes the outflow in our model. It is the
Newtonian analogue of the LTB spacetime of General Relativity
(e.g., Gromov et al. 2001)
.
A simpler model results if we can assume that the cluster
mass M0 makes up the majority of the mass M(r), and the flow
galaxies can be treated as test particles. Curiously enough, this
assumption makes the model more general in the sense that no
spherical symmetry is required now for the galaxy distribution
in the flow (not necessarily concentric shells around the
cluster). However, the gravity force produced by the cluster at the
distances of the flow is still isotropic. The model remains also
nonlinear: no restriction on the strength of the dynamical effects
is assumed.
The mass-point model may seem overly simple, because it is
known that the Virgo cluster contributes only 20% of the bright
galaxies within the volume up to the distance of the Local Group
(Tully 1982)
. However,
Tully & Mohayaee (2004)
conclude
from their numerical action method that the mass-to-luminosity
ratio of the E-galaxy rich Virgo cluster must be several times
higher (they give M/L ∼ 900) than in the surrounding region
(∼125) in order to produce the infall velocities observed close
to Virgo
(a qualitatively similar result was found by Teerikorpi
et al. 1992)
. Therefore, the model with most of the mass in and
close to the Virgo cluster could be a sufficient first
approximation.
In this simplified/generalized version, the mass M(r) = M
is constant in Eq. (4), which shows that antigravity and gravity
balance each other at the distance R = RZG, where
3M
8πρΛ
In this limit, FE/|FN| → ∞, and the gravity of the cluster
is practically negligible compared to the dark energy
antigravity. As a result, the flow asymptotically acquires the linear
velocity-distance relation with the constant Hubble factor
expressed via the dark energy density only:
HΛ =
8πG
3
ρΛ
1/2
At large distances (R RZG), the trajectories with E(r) = Emin
approach the asymptotic given by Eq. (11).
Another special value of the mechanical energy is E(r) = 0,
corresponding to parabolic motion. In this case,
V (r = ∞, t0) → HΛR(r, t0).
The flow trajectories converge to the asymptotic of Eq. (11) in
general independently of the “boundary/initial conditions” near
the cluster
(see Fig. 2 in Teerikorpi & Chernin 2010)
. In
particular, strong perturbations at small distances in the flow are
compatible with its regularity at larger distances. No tuning of
the model parameters is required for this – the only significant
quantity is the zero-gravity radius RZG determined by the cluster
mass and the DE density.
Thus, dark energy dominates the dynamics of the
Virgocentric flow on its entire observed extension from 12 to at
least 30 Mpc. Because of the strong antigravity, the flow proves
to be rather regular, acquiring the linear Hubble relation at
distances >15 Mpc from the cluster center. This major conclusion
∼
from the model also resolves the Hubble-Sandage paradox on
the scale of ∼10 Mpc, in harmony with our general approach to
the old puzzle (Sect. 1).
4. Discussion
We discuss here some specific features and implications of the
dark energy domination in the Virgocentric flow.
4.1. The Einstein-Straus radius
The mass of dark matter and baryons in the Virgo Cluster,
M = (0.6−1.2) × 1015 M , was initially collected from the
volume which at the present epoch has the radius RES =
[ 4πρ3MM(t0) ]1/3 = 15−19 Mpc, where ρM(t0) is the mean
matter density in the Universe. The quantity RES is known as the
Einstein-Straus radius in the vacuole model. In the presence
of dark energy
(Chernin et al. 2006; Teerikorpi et al. 2008)
,
RES = [ 2ρΛ ]1/3RZG = 1.7 RZG. Within the E-S vacuole, the
ρM(t0)
spherical layer between the radii R0 and RES does not contain
any mass. Therefore the approximation of test particles adopted
in the model of Sect. 3 works well in this layer. Beyond the
distance RES RMW the model needs modification, but remains
qualitatively correct up to the distances of 25−30 Mpc. However,
because 2ρΛ > ρM(t0), the antigravity of dark energy dominates
in the entire area of the Virgocentric flow in Fig. 1.
(10)
(11)
(12)
4.2. Two special trajectories
Equation (5) shows that the flow has zero velocity at R = RZG
R0, if the total energy of the shell or an individual galaxy
3 GM
E(r) = Emin = − 2 RZG ·
It is easy to see that Emin is the minimal energy needed for a
particle to escape from the potential well of the cluster. Equation (5)
with E(r) = Emin gives the corresponding velocity Vmin(R) of the
flow at different distances in the flow:
Vmin = HΛR 1 + 2
RZG 3
R
− 3
RZG
R
.
RZG
R
.
The two special trajectories with E(r) = Emin and E(r) = 0
(see Fig. 1) converge to the asymptotic of Eq. (11) at large
distances – in agreement with the results of Sect. 3.
It is quite interesting that the most accurate data on the flow
(from the TRGB and Cepheids; dots in Fig. 1) show a low
dispersion around the line V = HVR, and the corresponding
trajectories occupy the area just between the curves Vmin(R) and Vpar(R).
The rather wide spread of the other data in Fig. 1 is mostly due
to observational errors
(Karachentsev & Nasonova 2010 –
especially their Fig. 7b)
. Still, the restriction Emin ≤ E ≤ 0 puts an
upper limit on the value |E| in Eq. (5), and because of this, the
asymptotic of Eq. (11) may be reached more easily in the flow.
It is also significant that the Local Group outflow has the
same feature: the galaxies prefer the area between Vmin and Vpar
in the Hubble diagram of Fig. 2
(cf. Teerikorpi et al. 2008)
.
This is another similarity aspect between the Virgo system and
the Local system (Sect. 2). Mathematically, the similarity
reflects that Eqs. (13) and (14) have exactly the same view for
both systems in the dimensionless form y(x) = 0 with y =
V/(HΛRZG), x = R/RZG; the scale factor is again about 10.
This similarity suggests that the condition Emin ≤ E ≤ 0 has
more general significance; it may reflect the kinematical state of
the flow at the early formation epoch. For example, V > Vmin
follows, if the galaxies of the flow initially escaped from the
cluster
(Chernin et al. 2004, 2007c)
.
4.3. Comparison with cosmology
The zero-gravity radius RZG is a local spatial counterpart of the
redshift zΛ 0.7 at which the gravity and the antigravity
balance each other in the Universe as a whole. Globally, the balance
takes place only at one moment t(zΛ) in the entire co-moving
space, while the local gravity-antigravity balance exists during
the whole life-time of the system since its formation, but only
on the sphere of the radius RZG.
The Virgocentric flow should not be identified with the
global cosmological expansion. The Virgo system lies deeply
inside the cosmic cell of uniformity and does not “know” about
the universe of the horizon scales. In the early epoch of weak
protogalactic perturbations, the dark matter and baryons of the
system participated in the cosmological expansion. But later, at
the epoch of nonlinear perturbations, the matter separated from
the expansion and underwent violent evolution. Contrary to that,
the unperturbed flow at horizon-scale distances was not
similarly affected, and up to now the initial isotropy of its
expansive motion and the uniformity of its matter distribution are
conserved. Therefore, it is mysterious that the global Hubble
constant 72 km s−1 Mpc−1 is so close to the local expansion rate
65 km s−1 Mpc−1 in the Virgocentric flow (two beams in Fig. 1).
This is one aspect of the Hubble-Sandage paradox (Sect. 1).
Trying to find an explanation, we note that the Friedmann
equation for the scale factor a(t) has much the same form as the
local energy conservation Eq. (5):
(13)
a˙ (t)2 =
The constant C = 43π ρM(t)a(t)3 has the dimension of mass, and
ρM(t) is the density of matter (dark and baryonic); the energy
constant B = 0 in the standard flat cosmology. The cosmological
Hubble factor comes from this equation:
H(t) =
a˙(t)
a
=
8πG
3
1/2
(ρM(t) + ρΛ)
= HΛ 1 +
ρm
2ρΛ
1/2
· (16)
When time goes to infinity, the matter density drops to zero, the
dark energy becomes dominant, and the cosmological Hubble
factor becomes time-independent:
(17)
(18)
ΩM
ΩΛ
1/2
H(t) → HΛ =
8πG
3
ρΛ
1/2
which is equal to the empirical (WMAP) value.
As we see, the Virgocentric flow and the global
cosmological expansion have a common asymptotic in the limit of dark
energy domination: for the local flow, this is the asymptotic in
space (large distances from the cluster center), and for the global
flow, this is the asymptotic in time (a future of the Universe).
If the asymptotic is approached now both globally and locally,
we would have H HΛ on the largest scales and HV HΛ in
the Virgocentric flow.
In their present states, both flows are near the asymptotic,
though they do not quite reach it. That is why H, HV , and HΛ
have similar values. The flows are not directly related to each
other; but, they “feel” the same uniform dark energy
background, and in this way both are controlled by the antigravity
force, which is equally effective near the cosmic horizon and
at the periphery of the Virgocentric flow. As a result, the local
flow at its large distances looks much like a “part” of the global
horizon-distance flow.
The near coincidence of the values H, HV, and HΛ explained
by our model clarifies the Hubble-Sandage paradox (Sect. 1) for
the Virgocentric flow: asymptotically, the flow reaches not only
the linear Hubble law (Sect. 3), but also acquires the expansion
rate HV, which approaches the global rate when the measuring
accuracy is improved.
4.4. Probing local dark energy
The global density of dark energy can be derived if the
redshift zΛ is found at which the gravity of matter is exactly
balanced by the dark energy antigravity, and also the present matter
density is known. The same logic works in local studies: just
find the zero-gravity radius RZG from the outflow observations
and the mass of the local system.
We may now restrict the value of RZG using the diagram of
Fig. 1. Because the zero-gravity surface lies outside the cluster
volume, it should be RZG > R0 6−7 Mpc. On the other hand,
the linear velocity-distance relation, with the Hubble ratio close
to H, is clearly seen from a distance of about 15 Mpc. This
suggests that RZG < 15 Mpc. The argument that the zero-gravity
surface is located below the point where the local flow reaches
the global expansion rate gained support from the calculations by
Teerikorpi & Chernin (2010)
. As for the cluster mass, we adopt
the range M = (0.6−1.2) × 1015 M (Sect. 2). Then Eq. (6) in the
form ρx = M/( 83π R3ZG) directly leads to robust upper and lower
limits to the unknown local dark energy density ρx:
0.2 < ρx < 5 × 10−29 g cm−3.
Indeed, the upper limit obtained from the cluster size (RZG)
is conservative. We note that the theoretical minimum velocity
curve in Fig. 1 would be shifted up to the velocity-distance
relation defined by the TRGB and Cepheid distances, if one takes
ρx = 1.4 × 10−29 g cm−3. This would be a more realistic upper
limit, and near the limit similarly obtained in
Teerikorpi et al.
(2008)
using the Local Group and the M 81 group. Thus the
local dark energy density is near the global value or may be the
same; ρΛ = 0.73 × 10−29 g cm−3 lies comfortably in the range of
Eq. (19). Similar estimates for the dark energy come from
studies of the Local Group, M 81 group and Cen A group
(Chernin
et al. 2007a,b, 2009)
.
We may now argue reversely: let us assume that the density
of dark energy is known from cosmological observations and
the zero-gravity radius RZG is between 6 and 15 Mpc. Then we
may independently estimate the mass of the Virgo Cluster: M =
(0.3−4) × 1015 M . The value of the cluster mass adopted earlier
is within this interval.
(19)
4.5. The origin of the local flows
The model of Sect. 3 describes the present structure of
the Virgocentric flow and also predicts its future evolution.
However, it says almost nothing about the initial state of the
flow. Our discussion suggests that the Virgo cluster and the
outflow form a system with a common origin and evolutionary
history. Its formation was most probably caused by complex linear
and nonlinear processes (collisions and merging of galaxies and
protogalactic units, violent relaxation of the system in its
selfgravity field, etc.). Similar processes are expected in the early
history of the Local Group and other systems with the
twocomponent structure. Their basic features might be recognized
from big N-body simulations, like the Millennium Simulation
(Li & White 2008)
. We leave it for a later study to clarify the
way in which the galaxies of local flows gain their initial
expansion velocities.
5. Conclusions
The recently published systematic and most accurate data on the
velocities and distances of galaxies in the Virgo cluster and its
environment
(Karachentsev & Nasonova 2010)
shed light on the
physics behind the observed properties of the cluster and the
flow. We find that
1. The cluster and the flow can be treated together as a physical
system embedded in the uniform dark energy background;
this is adopted in our analytical nonlinear model.
2. The nonlinear interplay between the gravity produced by the
cluster mass (dark matter + baryons) and the antigravity of
the dark energy is the major dynamical factor determining
the kinematic structure of the system and controlling its
evolution; our model enables one to describe this in a simple
(exact or approximate) quantitative way.
3. The key physical parameter of the system is its zero-gravity
radius, RZG = 9−11 Mpc, below which the gravity
dominates the quasi-stationary bound cluster, while the
antigravity dominates the expanding Virgocentric flow.
4. The dark energy antigravity brings order and regularity to the
Virgocentric flow and the flow acquires the linear
velocitydistance relation at the Virgocentric distances R > 15 Mpc.
Thus, the Hubble-Sandage paradox is understood on the
cluster scale.
5. In the velocity-distance diagram, the zero-gravity radius is
near to or somewhat larger than the zero-velocity radius R0 =
5.0−7.5 Mpc, as determined by
Karachentsev & Nasonova
(2010)
.
6. With the value of RZG > R0 found from the diagram, one
∼
may estimate the mass of the Virgo cluster, using the known
global density of dark energy: M = (0.3−4) × 1015 M .
7. With the same value of RZG, one may estimate the local
density of the dark energy in the Virgo system, if the mass of
the cluster is considered known (from the virial or the
zerovelocity method): ρΛ = (0.2−5) × 10−29 g cm−3.
8. The two-component Virgo system is similar to the Local
system formed by the Local Group together with the local
outflow around it. In the velocity-distance diagram, the Virgo
system reproduces the Local system with the scale factor of
about 10. A dynamical similarity is seen in that the
zerogravity radius for the Virgo system is about 10 times larger
than for the Local system.
9. The similarity suggests that a universal two-component
grand design may exist with a quasi-stationary bound
central component and an expanding outflow around it on the
spatial scales of 1−30 Mpc. The phase and dynamical
structure of the real two-component systems reflects the nonlinear
gravity-antigravity interplay with dark energy domination in
the flow component.
Acknowledgements. A.C., V.D., and L.D. thank the RFBR for partial support
via the grant 10-02-00178. We also thank the anonymous referee for useful
comments.
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